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itemtype="http://schema.org/Person"><meta itemprop="name" content="阮行止"><meta itemprop="image" content="https://pic1.zhimg.com/2c7b78a58e2b3c11931f74e00099f4b7_is.jpg"><meta itemprop="url" content="https://www.zhihu.com/people/ruan-xing-zhi"><meta itemprop="zhihu:followerCount" content="3839"><span class="UserLink AuthorInfo-avatarWrapper"><div class="Popover"><div id="Popover11-toggle" aria-haspopup="true" aria-expanded="false" aria-owns="Popover11-content"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/ruan-xing-zhi"><img class="Avatar AuthorInfo-avatar" width="38" height="38" src="./Dynamic Programming_files/2c7b78a58e2b3c11931f74e00099f4b7_xs.jpg" srcset="https://pic1.zhimg.com/2c7b78a58e2b3c11931f74e00099f4b7_l.jpg 2x" alt="阮行止" data-evernote-lazy-src="https://pic1.zhimg.com/2c7b78a58e2b3c11931f74e00099f4b7_l.jpg"></a></div></div></span><div class="AuthorInfo-content"><div class="AuthorInfo-head"><span class="UserLink AuthorInfo-name"><div class="Popover"><div id="Popover12-toggle" aria-haspopup="true" aria-expanded="false" aria-owns="Popover12-content"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/ruan-xing-zhi">阮行止</a></div></div><a class="UserLink-badge" data-tooltip="已认证的个人" href="https://www.zhihu.com/question/48510028" target="_blank"><span style="display: inline-flex; align-items: center;">​<svg class="Zi Zi--BadgeCert" fill="currentColor" viewBox="0 0 24 24" width="18" height="18"><g fill="none" fill-rule="evenodd"><path fill="#0F88EB" d="M2.64 13.39c1.068.895 1.808 2.733 1.66 4.113l.022-.196c-.147 1.384.856 2.4 2.24 2.278l-.198.016c1.387-.122 3.21.655 4.083 1.734l-.125-.154c.876 1.084 2.304 1.092 3.195.027l-.127.152c.895-1.068 2.733-1.808 4.113-1.66l-.198-.022c1.386.147 2.402-.856 2.279-2.238l.017.197c-.122-1.388.655-3.212 1.734-4.084l-.154.125c1.083-.876 1.092-2.304.027-3.195l.152.127c-1.068-.895-1.808-2.732-1.66-4.113l-.022.198c.147-1.386-.856-2.4-2.24-2.279l.198-.017c-1.387.123-3.21-.654-4.083-1.733l.125.153c-.876-1.083-2.304-1.092-3.195-.027l.127-.152c-.895 1.068-2.733 1.808-4.113 1.662l.198.02c-1.386-.147-2.4.857-2.279 2.24L4.4 6.363c.122 1.387-.655 3.21-1.734 4.084l.154-.126c-1.083.878-1.092 2.304-.027 3.195l-.152-.127z"></path><path fill="#FFF" d="M9.78 15.728l-2.633-2.999s-.458-.705.242-1.362c.7-.657 1.328-.219 1.328-.219l1.953 2.132 4.696-4.931s.663-.348 1.299.198c.636.545.27 1.382.27 1.382s-3.466 3.858-5.376 5.782c-.98.93-1.778.017-1.778.017z"></path></g></svg></span></a></span></div><div class="AuthorInfo-detail"><div class="AuthorInfo-badge"><div class="AuthorInfo-badgeText">上海洛谷网络科技有限公司 讲师</div></div></div></div></div></div><meta itemprop="image"><meta itemprop="upvoteCount" content="5668"><meta itemprop="url" content="https://www.zhihu.com/question/23995189/answer/613096905"><meta itemprop="dateCreated" content="2019-03-03T08:40:06.000Z"><meta itemprop="dateModified" content="2019-03-03T09:09:44.000Z"><meta itemprop="commentCount" content="164"><div class="RichContent RichContent--unescapable"><div class="RichContent-inner"><span class="RichText ztext CopyrightRichText-richText" itemprop="text" ecommerce="[object Object]"><h2 class="">0. intro</h2><p class="">　　很有意思的问题。以往见过许多教材，对动态规划（DP）的引入属于“奉天承运，皇帝诏曰”式：不给出一点引入，见面即拿出一大堆公式吓人；学生则死啃书本，然后突然顿悟。针对入门者的教材不应该是这样的。恰好我给入门者讲过四次DP入门，迭代出了一套比较靠谱的教学方法，所以今天跑过来献丑。</p><p class="">　　现在，我们试着自己来一步步“重新发明”DP。</p><h2 class="">1. 从一个生活问题谈起</h2><p class="">　　先来看看生活中经常遇到的事吧——假设您是个土豪，身上带了足够的1、5、10、20、50、100元面值的钞票。现在您的目标是凑出某个金额w，<b class="">需要用到尽量少的钞票。</b></p><p class="">　　依据生活经验，我们显然可以采取这样的策略：能用100的就尽量用100的，否则尽量用50的……依次类推。在这种策略下，666=6×100+1×50+1×10+1×5+1×1，共使用了10张钞票。</p><p class="">　　这种策略称为“<b class="">贪心</b>”：假设我们面对的局面是“需要凑出w”，<b class="">贪心策略会<u class="">尽快</u>让w变得更小</b>。能让w少100就尽量让它少100，这样我们接下来面对的局面就是凑出w-100。长期的生活经验表明，贪心策略是正确的。</p><p class="">　　但是，如果我们换一组钞票的面值，贪心策略就也许不成立了。如果一个奇葩国家的钞票面额分别是1、5、11，那么我们在凑出15的时候，贪心策略会出错：<br>　　15=1×11+4×1    （贪心策略使用了5张钞票）<br>　　15=3×5               （正确的策略，只用3张钞票）<br>　　为什么会这样呢？贪心策略错在了哪里？</p><p class=""><b class="">　　鼠目寸光。</b><br>　　刚刚已经说过，贪心策略的纲领是：“尽量使接下来面对的w更小”。这样，贪心策略在w=15的局面时，会优先使用11来把w降到4；但是在这个问题中，凑出4的代价是很高的，必须使用4×1。如果使用了5，w会降为10，虽然没有4那么小，但是凑出10只需要两张5元。<br>　　在这里我们发现，贪心是一种<b class="">只考虑眼前情况</b>的策略。</p><p class="">　　那么，现在我们怎样才能避免鼠目寸光呢？</p><p class="">　　如果直接暴力枚举凑出w的方案，明显复杂度过高。太多种方法可以凑出w了，枚举它们的时间是不可承受的。我们现在来尝试找一下性质。</p><p class="ztext-empty-paragraph"><br></p><p class="">　　重新分析刚刚的例子。w=15时，我们如果取11，接下来就面对w=4的情况；如果取5，则接下来面对w=10的情况。我们发现这些问题都有相同的形式：“给定w，凑出w所用的最少钞票是多少张？”接下来，我们用f(n)来表示“凑出n所需的最少钞票数量”。</p><p class="">　　那么，如果我们取了11，最后的代价（用掉的钞票总数）是多少呢？<br>　　明显<img src="./Dynamic Programming_files/equation.svg" alt="[公式]" eeimg="1" data-formula="\text{cost} = f(4) + 1 = 4 + 1 = 5" class="" width="251" height="26"> ，它的意义是：利用11来凑出15，付出的代价等于f(4)加上自己这一张钞票。现在我们暂时不管f(4)怎么求出来。<br>　　依次类推，马上可以知道：如果我们用5来凑出15，cost就是<img src="./Dynamic Programming_files/equation(1).svg" alt="[公式]" eeimg="1" data-formula="f(10) + 1 = 2 + 1 = 3" class="" width="197" height="26"> 。</p><p class="">　　那么，现在w=15的时候，我们该取那种钞票呢？<b class="">当然是各种方案中，cost值最低的那一个</b>！</p><p class="">　　- 取11：<img src="./Dynamic Programming_files/equation(2).svg" alt="[公式]" eeimg="1" data-formula="\text{cost}=f(4)+1=4+1=5" class="" width="251" height="26"><br>　　- 取5：  <img src="./Dynamic Programming_files/equation(3).svg" alt="[公式]" eeimg="1" data-formula="\text{cost}=f(10)+1=2+1=3" class="" width="261" height="26"><br>　　- 取1：  <img src="./Dynamic Programming_files/equation(4).svg" alt="[公式]" eeimg="1" data-formula="\text{cost}=f(14)+1=4+1=5" class="" width="261" height="26"></p><p class="">　　显而易见，cost值最低的是取5的方案。<b class="">我们通过上面三个式子，做出了正确的决策</b>！</p><p class="">　　这给了我们一个<b class="">至关重要</b>的启示—— <img src="./Dynamic Programming_files/equation(5).svg" alt="[公式]" eeimg="1" data-formula="f(n)" class="" width="40" height="25"> 只与 <img src="./Dynamic Programming_files/equation(6).svg" alt="[公式]" eeimg="1" data-formula="f(n-1),f(n-5),f(n-11)" class="" width="258" height="26"> 相关；更确切地说：</p><p class=""><img src="./Dynamic Programming_files/equation(7).svg" alt="[公式]" eeimg="1" data-formula="f(n)=\min\{f(n-1),f(n-5),f(n-11)\}+1" class="" width="418" height="26"></p><p class="">　　这个式子是非常激动人心的。我们要求出f(n)，只需要求出几个更小的f值；既然如此，我们从小到大把所有的f(i)求出来不就好了？注意一下边界情况即可。代码如下：</p><figure data-size="normal" class=""><noscript><img src="https://pic2.zhimg.com/50/v2-6a5ba74fb90968533ece429ed329c903_hd.jpg" data-rawwidth="1044" data-rawheight="440" data-size="normal" data-caption="" data-default-watermark-src="https://pic2.zhimg.com/50/v2-af8977bff8d8e7f41d72b2e0416c83d6_hd.jpg" class="origin_image zh-lightbox-thumb" width="1044" data-original="https://pic2.zhimg.com/v2-6a5ba74fb90968533ece429ed329c903_r.jpg"/></noscript><img src="./Dynamic Programming_files/v2-6a5ba74fb90968533ece429ed329c903_r.jpg" data-rawwidth="1044" data-rawheight="440" data-size="normal" data-caption="" data-default-watermark-src="https://pic2.zhimg.com/50/v2-af8977bff8d8e7f41d72b2e0416c83d6_hd.jpg" class="origin_image zh-lightbox-thumb lazy" width="1044" data-original="https://pic2.zhimg.com/v2-6a5ba74fb90968533ece429ed329c903_r.jpg" data-actualsrc="https://pic2.zhimg.com/50/v2-6a5ba74fb90968533ece429ed329c903_hd.jpg" data-evernote-lazy-src="https://pic2.zhimg.com/v2-6a5ba74fb90968533ece429ed329c903_r.jpg" height="276"></figure><p class="">　　我们以 <img src="./Dynamic Programming_files/equation(8).svg" alt="[公式]" eeimg="1" data-formula="O(n)" class="" width="45" height="26"> 的复杂度解决了这个问题。现在回过头来，我们看看它的原理：</p><p class="">　　- <img src="./Dynamic Programming_files/equation(5).svg" alt="[公式]" eeimg="1" data-formula="f(n)" class="" width="40" height="25"> 只与<img src="./Dynamic Programming_files/equation(6).svg" alt="[公式]" eeimg="1" data-formula="f(n-1),f(n-5),f(n-11)" class="" width="258" height="26">的<b class="">值</b>相关。<br>　　-  我们只关心 <img src="./Dynamic Programming_files/equation(9).svg" alt="[公式]" eeimg="1" data-formula="f(w)" class="" width="43" height="26"> 的<b class="">值</b>，不关心是怎么凑出w的。</p><p class="">　　这两个事实，保证了我们做法的正确性。它比起贪心策略，会分别算出取1、5、11的代价，从而做出一个正确决策，这样就避免掉了“鼠目寸光”！</p><p class="">　　它与暴力的区别在哪里？我们的暴力枚举了“使用的硬币”，然而这属于冗余信息。我们要的是答案，根本不关心这个答案是怎么凑出来的。譬如，要求出f(15)，只需要知道f(14),f(10),f(4)的值。<b class="">其他信息并不需要。</b>我们舍弃了冗余信息。我们只记录了对解决问题有帮助的信息——f(n).</p><p class="">　　我们能这样干，取决于问题的性质：求出f(n)，只需要知道几个更小的f(c)。<b class="">我们将求解f(c)称作求解f(n)的“子问题”。</b></p><p class="ztext-empty-paragraph"><br></p><p class=""><b class="">　　这就是DP</b>（动态规划，dynamic programming）.</p><p class=""><b class="">　　将一个问题拆成几个子问题，分别求解这些子问题，即可推断出大问题的解</b>。</p><blockquote class="">思考题：请稍微修改代码，输出我们凑出w的<b class="">方案</b>。</blockquote><h2 class="">2. 几个简单的概念</h2><p class="">【无后效性】</p><p class="">　　一旦f(n)确定，“我们如何凑出f(n)”就再也用不着了。</p><p class="">　　要求出f(15)，只需要知道f(14),f(10),f(4)的值，而f(14),f(10),f(4)是如何算出来的，对之后的问题没有影响。</p><p class=""><b class="">　　“未来与过去无关”，</b>这就是<b class="">无后效性</b>。</p><p class="">　　（严格定义：如果给定某一阶段的状态，则在这一阶段以后过程的发展不受这阶段以前各段状态的影响。）</p><p class="ztext-empty-paragraph"><br></p><p class="">【最优子结构】</p><p class="">　　回顾我们对f(n)的定义：我们记“凑出n所需的<b class="">最少</b>钞票数量”为f(n).</p><p class="">　　f(n)的定义就已经蕴含了“最优”。利用w=14,10,4的<b class="">最优</b>解，我们即可算出w=15的<b class="">最优</b>解。</p><p class="">　　大问题的<b class="">最优解</b>可以由小问题的<b class="">最优解</b>推出，这个性质叫做“最优子结构性质”。</p><p class="ztext-empty-paragraph"><br></p><p class="">　　引入这两个概念之后，我们如何判断一个问题能否使用DP解决呢？</p><p class="ztext-empty-paragraph"><br></p><p class=""><b class="">　　能将大问题拆成几个小问题，且满足无后效性、最优子结构性质。</b></p><h2 class=""><b class="">3. DP的典型应用：DAG最短路</b></h2><p class="">　　问题很简单：给定一个城市的地图，所有的道路都是单行道，而且不会构成环。每条道路都有过路费，问您从S点到T点花费的最少费用。</p><figure data-size="normal" class=""><noscript><img src="https://pic1.zhimg.com/50/v2-38e9a487997d2eea979097fbc9e9e674_hd.jpg" data-rawwidth="523" data-rawheight="251" data-size="normal" data-default-watermark-src="https://pic3.zhimg.com/50/v2-4d9f5b84e0d1f8aafe616a36b869a012_hd.jpg" class="origin_image zh-lightbox-thumb" width="523" data-original="https://pic1.zhimg.com/v2-38e9a487997d2eea979097fbc9e9e674_r.jpg"/></noscript><img src="./Dynamic Programming_files/v2-38e9a487997d2eea979097fbc9e9e674_r.jpg" data-rawwidth="523" data-rawheight="251" data-size="normal" data-default-watermark-src="https://pic3.zhimg.com/50/v2-4d9f5b84e0d1f8aafe616a36b869a012_hd.jpg" class="origin_image zh-lightbox-thumb lazy" width="523" data-original="https://pic1.zhimg.com/v2-38e9a487997d2eea979097fbc9e9e674_r.jpg" data-actualsrc="https://pic1.zhimg.com/50/v2-38e9a487997d2eea979097fbc9e9e674_hd.jpg" data-evernote-lazy-src="https://pic1.zhimg.com/v2-38e9a487997d2eea979097fbc9e9e674_r.jpg" height="251"><figcaption class="">一张地图。边上的数字表示过路费。</figcaption></figure><p class="">　　这个问题能用DP解决吗？我们先试着记从S到P的最少费用为f(P).<br>　　想要到T，要么经过C，要么经过D。从而<img src="./Dynamic Programming_files/equation(10).svg" alt="[公式]" eeimg="1" data-formula="f(T)=\min⁡\{f(C)+20,f(D)+10\}" class="" width="321" height="26">.</p><p class="">　　好像看起来可以DP。现在我们检验刚刚那两个性质：<br>　　- 无后效性：对于点P，一旦f(P)确定，以后就只关心f(P)的值，不关心怎么去的。<br>　　- 最优子结构：对于P，我们当然只关心到P的最小费用，即f(P)。如果我们从S走到T是 <img src="./Dynamic Programming_files/equation(11).svg" alt="[公式]" eeimg="1" data-formula="S \to P\to Q\to T" class="" width="158" height="23"> ，那肯定S走到Q的最优路径是 <img src="./Dynamic Programming_files/equation(12).svg" alt="[公式]" eeimg="1" data-formula="S\to P\to Q" class="" width="111" height="23"> 。对一条最优的路径而言，从S走到<b class="">沿途上所有的点（子问题）</b>的最优路径，都是这条大路的一部分。这个问题的最优子结构性质是显然的。</p><p class="">　　既然这两个性质都满足，那么本题可以DP。式子明显为：</p><p class=""><img src="./Dynamic Programming_files/equation(13).svg" alt="[公式]" eeimg="1" data-formula="f(P)=\min⁡\{f(R)+w_{R→P}\}" class="" width="254" height="26"></p><p class="">　　其中R为有路通到P的所有的点， <img src="./Dynamic Programming_files/equation(14).svg" alt="[公式]" eeimg="1" data-formula="w_{R→P}" class="" width="54" height="18"> 为R到P的过路费。</p><p class="">　　代码实现也很简单，拓扑排序即可。</p><h2 class="">4. 对DP原理的一点讨论</h2><p class="">【DP的核心思想】</p><p class="">　　DP为什么会快？<br>　　无论是DP还是暴力，我们的算法都是在<b class="">可能解空间</b>内，寻找<b class="">最优解</b>。</p><p class="">　　来看钞票问题。暴力做法是枚举所有的可能解，这是最大的可能解空间。<br>　　DP是枚举<b class="">有希望成为答案的解</b>。这个空间比暴力的小得多。</p><p class="">　　也就是说：DP自带剪枝。</p><p class="">　　DP舍弃了一大堆不可能成为最优解的答案。譬如：<br>　　15 = 5+5+5 被考虑了。<br>　　15 = 5+5+1+1+1+1+1 从来没有考虑过，因为这不可能成为最优解。</p><p class="">　　从而我们可以得到DP的核心思想：<b class="">尽量缩小可能解空间。</b></p><p class="">　　在暴力算法中，可能解空间往往是指数级的大小；如果我们采用DP，那么有可能把解空间的大小降到多项式级。</p><p class="">　　一般来说，解空间越小，寻找解就越快。这样就完成了优化。</p><p class="ztext-empty-paragraph"><br></p><p class="">【DP的操作过程】</p><p class="">　　一言以蔽之：<b class="">大事化小，小事化了。</b></p><p class="">　　将一个大问题转化成几个小问题；<br>　　求解小问题；<br>　　推出大问题的解。</p><p class="">【如何设计DP算法】</p><p class="">　　下面介绍比较通用的设计DP算法的步骤。</p><p class="">　　首先，把我们面对的<b class="">局面</b>表示为x。这一步称为<b class="">设计状态</b>。<br>　　对于状态x，记我们要求出的答案(e.g. 最小费用)为f(x).我们的目标是求出f(T).<br><b class="">找出f(x)与哪些局面有关（记为p）</b>，写出一个式子（称为<b class="">状态转移方程</b>），通过f(p)来推出f(x).</p><p class="">【DP三连】</p><p class="">　　设计DP算法，往往可以遵循DP三连：</p><p class="">　　我是谁？  ——设计状态，表示局面<br>　　我从哪里来？<br>　　我要到哪里去？  ——设计转移</p><p class="">　　设计状态是DP的基础。接下来的设计转移，有两种方式：一种是考虑我从哪里来（本文之前提到的两个例子，都是在考虑“我从哪里来”）；另一种是考虑我到哪里去，这常见于求出f(x)之后，<b class="">更新能从x走到的一些解</b>。这种DP也是不少的，我们以后会遇到。</p><p class="">　　总而言之，“我从哪里来”和“我要到哪里去”只需要考虑清楚其中一个，就能设计出状态转移方程，从而写代码求解问题。前者又称pull型的转移，后者又称push型的转移。（这两个词是 <span><span class="UserLink"><div class="Popover"><div id="Popover19-toggle" aria-haspopup="true" aria-expanded="false" aria-owns="Popover19-content"><a class="UserLink-link" data-za-detail-view-element_name="User" target="_blank" href="https://www.zhihu.com/people/31585b1079521ef37391925a95a5d4ba" data-za-detail-view-id="1045">@阮止雨</a></div></div></span></span> 妹妹告诉我的，不知道源出处在哪）</p><blockquote class="">思考题：如何把钞票问题的代码改写成“我到哪里去”的形式？<br>提示：求出f(x)之后，更新f(x+1),f(x+5),f(x+11).</blockquote><h2 class="">5. 例题：最长上升子序列</h2><p class="">　　扯了这么多形而上的内容，还是做一道例题吧。</p><p class="">　　最长上升子序列（LIS）问题：给定长度为n的序列a，从a中抽取出一个子序列，这个子序列需要单调递增。问最长的上升子序列（LIS）的长度。<br>　　e.g. 1,5,3,4,6,9,7,8的LIS为1,3,4,6,7,8，长度为6。</p><p class="ztext-empty-paragraph"><br></p><p class="">　　如何设计状态（我是谁）？</p><p class="">　　我们记 <img src="./Dynamic Programming_files/equation(15).svg" alt="[公式]" eeimg="1" data-formula="f(x)" class="" width="40" height="26"> 为以 <img src="./Dynamic Programming_files/equation(16).svg" alt="[公式]" eeimg="1" data-formula="a_x" class="" width="22" height="18"> 结尾的LIS长度，那么答案就是 <img src="./Dynamic Programming_files/equation(17).svg" alt="[公式]" eeimg="1" data-formula="\max\{f(x)\}" class="" width="100" height="26"> .</p><p class="ztext-empty-paragraph"><br></p><p class="">　　状态x从哪里推过来（我从哪里来）？</p><p class="">　　考虑比x小的每一个p：如果 <img src="./Dynamic Programming_files/equation(18).svg" alt="[公式]" eeimg="1" data-formula="a_x&gt;a_p" class="" width="70" height="23"> ，那么f(x)可以取f(p)+1.<br>　　解释：我们把 <img src="./Dynamic Programming_files/equation(16).svg" alt="[公式]" eeimg="1" data-formula="a_x" class="" width="22" height="18"> 接在 <img src="./Dynamic Programming_files/equation(19).svg" alt="[公式]" eeimg="1" data-formula="a_p" class="" width="21" height="21"> 的后面，肯定能构造一个以 <img src="./Dynamic Programming_files/equation(16).svg" alt="[公式]" eeimg="1" data-formula="a_x" class="" width="22" height="18"> 结尾的上升子序列，长度比以 <img src="./Dynamic Programming_files/equation(19).svg" alt="[公式]" eeimg="1" data-formula="a_p" class="" width="21" height="21"> 结尾的LIS大1.那么，我们可以写出状态转移方程了：</p><p class=""><img src="./Dynamic Programming_files/equation(20).svg" alt="[公式]" eeimg="1" data-formula="f(x)=\max_{p&lt;x , a_p&lt;a_x }⁡\{f(p)\}+1" class="" width="241" height="39"></p><p class="">　　至此解决问题。两层for循环，复杂度 <img src="./Dynamic Programming_files/equation(21).svg" alt="[公式]" eeimg="1" data-formula="O(n^2)" class="" width="54" height="28"> .</p><figure data-size="normal" class=""><noscript><img src="https://pic3.zhimg.com/50/v2-73ea19922aaac11c15dff9146a5c5b41_hd.jpg" data-rawwidth="883" data-rawheight="455" data-size="normal" data-caption="" data-default-watermark-src="https://pic2.zhimg.com/50/v2-ce272e9f786ef6377691df709264711a_hd.jpg" class="origin_image zh-lightbox-thumb" width="883" data-original="https://pic3.zhimg.com/v2-73ea19922aaac11c15dff9146a5c5b41_r.jpg"/></noscript><img src="./Dynamic Programming_files/v2-73ea19922aaac11c15dff9146a5c5b41_r.jpg" data-rawwidth="883" data-rawheight="455" data-size="normal" data-caption="" data-default-watermark-src="https://pic2.zhimg.com/50/v2-ce272e9f786ef6377691df709264711a_hd.jpg" class="origin_image zh-lightbox-thumb lazy" width="883" data-original="https://pic3.zhimg.com/v2-73ea19922aaac11c15dff9146a5c5b41_r.jpg" data-actualsrc="https://pic3.zhimg.com/50/v2-73ea19922aaac11c15dff9146a5c5b41_hd.jpg" data-evernote-lazy-src="https://pic3.zhimg.com/v2-73ea19922aaac11c15dff9146a5c5b41_r.jpg" height="337"></figure><p class="">　　从这三个例题中可以看出，DP是一种思想，一种“大事化小，小事化了”的思想。带着这种思想，DP将会成为我们解决问题的利器。</p><p class="">　　最后，我们一起念一遍DP三连吧——我是谁？我从哪里来？我要到哪里去？</p><h2 class="">6. 习题</h2><p class="">如果读者有兴趣，可以试着完成下面几个习题：</p><p class="">一、请采取一些优化手段，以 <img src="./Dynamic Programming_files/equation(22).svg" alt="[公式]" eeimg="1" data-formula="O(n\log n)" class="" width="91" height="26"> 的复杂度解决LIS问题。</p><p class="">提示：可以参考这篇博客 <a href="https://link.zhihu.com/?target=https%3A//pks-loving.blog.luogu.org/junior-dynamic-programming-dong-tai-gui-hua-chu-bu-ge-zhong-zi-xu-lie" class="wrap external" target="_blank" rel="nofollow noreferrer" data-za-detail-view-id="1043">Junior Dynamic Programming--动态规划初步·各种子序列问题</a></p><p class="">二、“按顺序递推”和“记忆化搜索”是实现DP的两种方式。请查阅资料，简单描述“记忆化搜索”是什么。并采用记忆化搜索写出钞票问题的代码，然后完成<a href="https://link.zhihu.com/?target=https%3A//www.luogu.org/problemnew/show/P1541" class="wrap external" target="_blank" rel="nofollow noreferrer" data-za-detail-view-id="1043">P1541 乌龟棋 - 洛谷</a> 。</p><p class="">三、01背包问题是一种常见的DP模型。请完成<a href="https://link.zhihu.com/?target=https%3A//www.luogu.org/problemnew/show/P1048" class="wrap external" target="_blank" rel="nofollow noreferrer" data-za-detail-view-id="1043">P1048 采药 - 洛谷</a>。</p><p class="ztext-empty-paragraph"><br></p><p class="">谢谢您看完本文 ⁄(⁄ ⁄ ⁄ω⁄ ⁄ ⁄)⁄</p><p class="">2019.3.3</p></span></div><div><div class="ContentItem-time"><a target="_blank" href="https://www.zhihu.com/question/23995189/answer/613096905" class=""><span data-tooltip="发布于 2019-03-03 16:40">编辑于 2019-03-03</span></a></div></div></div></div></div></div></div></div></div></div></main><div class="CornerButtons"><div class="CornerAnimayedFlex"><button data-tooltip="建议反馈" 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从一个生活问题谈起\u003C\u002Fh2\u003E\u003Cp\u003E　　先来看看生活中经常遇到的事吧——假设您是个土豪，身上带了足够的1、5、10、20、50、100元面值的钞票。现在您的目标是凑出某个金额w，\u003Cb\u003E需要用到尽量少的钞票。\u003C\u002Fb\u003E\u003C\u002Fp\u003E\u003Cp\u003E　　依据生活经验，我们显然可以采取这样的策略：能用100的就尽量用100的，否则尽量用50的……依次类推。在这种策略下，666=6×100+1×50+1×10+1×5+1×1，共使用了10张钞票。\u003C\u002Fp\u003E\u003Cp\u003E　　这种策略称为“\u003Cb\u003E贪心\u003C\u002Fb\u003E”：假设我们面对的局面是“需要凑出w”，\u003Cb\u003E贪心策略会\u003Cu\u003E尽快\u003C\u002Fu\u003E让w变得更小\u003C\u002Fb\u003E。能让w少100就尽量让它少100，这样我们接下来面对的局面就是凑出w-100。长期的生活经验表明，贪心策略是正确的。\u003C\u002Fp\u003E\u003Cp\u003E　　但是，如果我们换一组钞票的面值，贪心策略就也许不成立了。如果一个奇葩国家的钞票面额分别是1、5、11，那么我们在凑出15的时候，贪心策略会出错：\u003Cbr\u002F\u003E　　15=1×11+4×1    （贪心策略使用了5张钞票）\u003Cbr\u002F\u003E　　15=3×5               （正确的策略，只用3张钞票）\u003Cbr\u002F\u003E　　为什么会这样呢？贪心策略错在了哪里？\u003C\u002Fp\u003E\u003Cp\u003E\u003Cb\u003E　　鼠目寸光。\u003C\u002Fb\u003E\u003Cbr\u002F\u003E　　刚刚已经说过，贪心策略的纲领是：“尽量使接下来面对的w更小”。这样，贪心策略在w=15的局面时，会优先使用11来把w降到4；但是在这个问题中，凑出4的代价是很高的，必须使用4×1。如果使用了5，w会降为10，虽然没有4那么小，但是凑出10只需要两张5元。\u003Cbr\u002F\u003E　　在这里我们发现，贪心是一种\u003Cb\u003E只考虑眼前情况\u003C\u002Fb\u003E的策略。\u003C\u002Fp\u003E\u003Cp\u003E　　那么，现在我们怎样才能避免鼠目寸光呢？\u003C\u002Fp\u003E\u003Cp\u003E　　如果直接暴力枚举凑出w的方案，明显复杂度过高。太多种方法可以凑出w了，枚举它们的时间是不可承受的。我们现在来尝试找一下性质。\u003C\u002Fp\u003E\u003Cp class=\"ztext-empty-paragraph\"\u003E\u003Cbr\u002F\u003E\u003C\u002Fp\u003E\u003Cp\u003E　　重新分析刚刚的例子。w=15时，我们如果取11，接下来就面对w=4的情况；如果取5，则接下来面对w=10的情况。我们发现这些问题都有相同的形式：“给定w，凑出w所用的最少钞票是多少张？”接下来，我们用f(n)来表示“凑出n所需的最少钞票数量”。\u003C\u002Fp\u003E\u003Cp\u003E　　那么，如果我们取了11，最后的代价（用掉的钞票总数）是多少呢？\u003Cbr\u002F\u003E　　明显\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Ctext%7Bcost%7D+%3D+f%284%29+%2B+1+%3D+4+%2B+1+%3D+5\" alt=\"\\text{cost} = f(4) + 1 = 4 + 1 = 5\" eeimg=\"1\"\u002F\u003E ，它的意义是：利用11来凑出15，付出的代价等于f(4)加上自己这一张钞票。现在我们暂时不管f(4)怎么求出来。\u003Cbr\u002F\u003E　　依次类推，马上可以知道：如果我们用5来凑出15，cost就是\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f%2810%29+%2B+1+%3D+2+%2B+1+%3D+3\" alt=\"f(10) + 1 = 2 + 1 = 3\" eeimg=\"1\"\u002F\u003E 。\u003C\u002Fp\u003E\u003Cp\u003E　　那么，现在w=15的时候，我们该取那种钞票呢？\u003Cb\u003E当然是各种方案中，cost值最低的那一个\u003C\u002Fb\u003E！\u003C\u002Fp\u003E\u003Cp\u003E　　- 取11：\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Ctext%7Bcost%7D%3Df%284%29%2B1%3D4%2B1%3D5\" alt=\"\\text{cost}=f(4)+1=4+1=5\" eeimg=\"1\"\u002F\u003E\u003Cbr\u002F\u003E　　- 取5：  \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Ctext%7Bcost%7D%3Df%2810%29%2B1%3D2%2B1%3D3\" alt=\"\\text{cost}=f(10)+1=2+1=3\" eeimg=\"1\"\u002F\u003E\u003Cbr\u002F\u003E　　- 取1：  \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Ctext%7Bcost%7D%3Df%2814%29%2B1%3D4%2B1%3D5\" alt=\"\\text{cost}=f(14)+1=4+1=5\" eeimg=\"1\"\u002F\u003E\u003C\u002Fp\u003E\u003Cp\u003E　　显而易见，cost值最低的是取5的方案。\u003Cb\u003E我们通过上面三个式子，做出了正确的决策\u003C\u002Fb\u003E！\u003C\u002Fp\u003E\u003Cp\u003E　　这给了我们一个\u003Cb\u003E至关重要\u003C\u002Fb\u003E的启示—— \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f%28n%29\" alt=\"f(n)\" eeimg=\"1\"\u002F\u003E 只与 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f%28n-1%29%2Cf%28n-5%29%2Cf%28n-11%29\" alt=\"f(n-1),f(n-5),f(n-11)\" eeimg=\"1\"\u002F\u003E 相关；更确切地说：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f%28n%29%3D%5Cmin%5C%7Bf%28n-1%29%2Cf%28n-5%29%2Cf%28n-11%29%5C%7D%2B1\" alt=\"f(n)=\\min\\{f(n-1),f(n-5),f(n-11)\\}+1\" eeimg=\"1\"\u002F\u003E\u003C\u002Fp\u003E\u003Cp\u003E　　这个式子是非常激动人心的。我们要求出f(n)，只需要求出几个更小的f值；既然如此，我们从小到大把所有的f(i)求出来不就好了？注意一下边界情况即可。代码如下：\u003C\u002Fp\u003E\u003Cfigure data-size=\"normal\"\u003E\u003Cnoscript\u003E\u003Cimg src=\"https:\u002F\u002Fpic2.zhimg.com\u002F50\u002Fv2-6a5ba74fb90968533ece429ed329c903_hd.jpg\" data-rawwidth=\"1044\" data-rawheight=\"440\" data-size=\"normal\" data-caption=\"\" data-default-watermark-src=\"https:\u002F\u002Fpic2.zhimg.com\u002F50\u002Fv2-af8977bff8d8e7f41d72b2e0416c83d6_hd.jpg\" class=\"origin_image zh-lightbox-thumb\" width=\"1044\" data-original=\"https:\u002F\u002Fpic2.zhimg.com\u002Fv2-6a5ba74fb90968533ece429ed329c903_r.jpg\"\u002F\u003E\u003C\u002Fnoscript\u003E\u003Cimg src=\"data:image\u002Fsvg+xml;utf8,&lt;svg xmlns=&#39;http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg&#39; width=&#39;1044&#39; height=&#39;440&#39;&gt;&lt;\u002Fsvg&gt;\" data-rawwidth=\"1044\" data-rawheight=\"440\" data-size=\"normal\" data-caption=\"\" data-default-watermark-src=\"https:\u002F\u002Fpic2.zhimg.com\u002F50\u002Fv2-af8977bff8d8e7f41d72b2e0416c83d6_hd.jpg\" class=\"origin_image zh-lightbox-thumb lazy\" width=\"1044\" data-original=\"https:\u002F\u002Fpic2.zhimg.com\u002Fv2-6a5ba74fb90968533ece429ed329c903_r.jpg\" data-actualsrc=\"https:\u002F\u002Fpic2.zhimg.com\u002F50\u002Fv2-6a5ba74fb90968533ece429ed329c903_hd.jpg\"\u002F\u003E\u003C\u002Ffigure\u003E\u003Cp\u003E　　我们以 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=O%28n%29\" alt=\"O(n)\" eeimg=\"1\"\u002F\u003E 的复杂度解决了这个问题。现在回过头来，我们看看它的原理：\u003C\u002Fp\u003E\u003Cp\u003E　　- \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f%28n%29\" alt=\"f(n)\" eeimg=\"1\"\u002F\u003E 只与\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f%28n-1%29%2Cf%28n-5%29%2Cf%28n-11%29\" alt=\"f(n-1),f(n-5),f(n-11)\" eeimg=\"1\"\u002F\u003E的\u003Cb\u003E值\u003C\u002Fb\u003E相关。\u003Cbr\u002F\u003E　　-  我们只关心 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f%28w%29\" alt=\"f(w)\" eeimg=\"1\"\u002F\u003E 的\u003Cb\u003E值\u003C\u002Fb\u003E，不关心是怎么凑出w的。\u003C\u002Fp\u003E\u003Cp\u003E　　这两个事实，保证了我们做法的正确性。它比起贪心策略，会分别算出取1、5、11的代价，从而做出一个正确决策，这样就避免掉了“鼠目寸光”！\u003C\u002Fp\u003E\u003Cp\u003E　　它与暴力的区别在哪里？我们的暴力枚举了“使用的硬币”，然而这属于冗余信息。我们要的是答案，根本不关心这个答案是怎么凑出来的。譬如，要求出f(15)，只需要知道f(14),f(10),f(4)的值。\u003Cb\u003E其他信息并不需要。\u003C\u002Fb\u003E我们舍弃了冗余信息。我们只记录了对解决问题有帮助的信息——f(n).\u003C\u002Fp\u003E\u003Cp\u003E　　我们能这样干，取决于问题的性质：求出f(n)，只需要知道几个更小的f(c)。\u003Cb\u003E我们将求解f(c)称作求解f(n)的“子问题”。\u003C\u002Fb\u003E\u003C\u002Fp\u003E\u003Cp class=\"ztext-empty-paragraph\"\u003E\u003Cbr\u002F\u003E\u003C\u002Fp\u003E\u003Cp\u003E\u003Cb\u003E　　这就是DP\u003C\u002Fb\u003E（动态规划，dynamic programming）.\u003C\u002Fp\u003E\u003Cp\u003E\u003Cb\u003E　　将一个问题拆成几个子问题，分别求解这些子问题，即可推断出大问题的解\u003C\u002Fb\u003E。\u003C\u002Fp\u003E\u003Cblockquote\u003E思考题：请稍微修改代码，输出我们凑出w的\u003Cb\u003E方案\u003C\u002Fb\u003E。\u003C\u002Fblockquote\u003E\u003Ch2\u003E2. 几个简单的概念\u003C\u002Fh2\u003E\u003Cp\u003E【无后效性】\u003C\u002Fp\u003E\u003Cp\u003E　　一旦f(n)确定，“我们如何凑出f(n)”就再也用不着了。\u003C\u002Fp\u003E\u003Cp\u003E　　要求出f(15)，只需要知道f(14),f(10),f(4)的值，而f(14),f(10),f(4)是如何算出来的，对之后的问题没有影响。\u003C\u002Fp\u003E\u003Cp\u003E\u003Cb\u003E　　“未来与过去无关”，\u003C\u002Fb\u003E这就是\u003Cb\u003E无后效性\u003C\u002Fb\u003E。\u003C\u002Fp\u003E\u003Cp\u003E　　（严格定义：如果给定某一阶段的状态，则在这一阶段以后过程的发展不受这阶段以前各段状态的影响。）\u003C\u002Fp\u003E\u003Cp class=\"ztext-empty-paragraph\"\u003E\u003Cbr\u002F\u003E\u003C\u002Fp\u003E\u003Cp\u003E【最优子结构】\u003C\u002Fp\u003E\u003Cp\u003E　　回顾我们对f(n)的定义：我们记“凑出n所需的\u003Cb\u003E最少\u003C\u002Fb\u003E钞票数量”为f(n).\u003C\u002Fp\u003E\u003Cp\u003E　　f(n)的定义就已经蕴含了“最优”。利用w=14,10,4的\u003Cb\u003E最优\u003C\u002Fb\u003E解，我们即可算出w=15的\u003Cb\u003E最优\u003C\u002Fb\u003E解。\u003C\u002Fp\u003E\u003Cp\u003E　　大问题的\u003Cb\u003E最优解\u003C\u002Fb\u003E可以由小问题的\u003Cb\u003E最优解\u003C\u002Fb\u003E推出，这个性质叫做“最优子结构性质”。\u003C\u002Fp\u003E\u003Cp class=\"ztext-empty-paragraph\"\u003E\u003Cbr\u002F\u003E\u003C\u002Fp\u003E\u003Cp\u003E　　引入这两个概念之后，我们如何判断一个问题能否使用DP解决呢？\u003C\u002Fp\u003E\u003Cp class=\"ztext-empty-paragraph\"\u003E\u003Cbr\u002F\u003E\u003C\u002Fp\u003E\u003Cp\u003E\u003Cb\u003E　　能将大问题拆成几个小问题，且满足无后效性、最优子结构性质。\u003C\u002Fb\u003E\u003C\u002Fp\u003E\u003Ch2\u003E\u003Cb\u003E3. DP的典型应用：DAG最短路\u003C\u002Fb\u003E\u003C\u002Fh2\u003E\u003Cp\u003E　　问题很简单：给定一个城市的地图，所有的道路都是单行道，而且不会构成环。每条道路都有过路费，问您从S点到T点花费的最少费用。\u003C\u002Fp\u003E\u003Cfigure data-size=\"normal\"\u003E\u003Cnoscript\u003E\u003Cimg src=\"https:\u002F\u002Fpic1.zhimg.com\u002F50\u002Fv2-38e9a487997d2eea979097fbc9e9e674_hd.jpg\" data-rawwidth=\"523\" data-rawheight=\"251\" data-size=\"normal\" data-default-watermark-src=\"https:\u002F\u002Fpic3.zhimg.com\u002F50\u002Fv2-4d9f5b84e0d1f8aafe616a36b869a012_hd.jpg\" class=\"origin_image zh-lightbox-thumb\" width=\"523\" data-original=\"https:\u002F\u002Fpic1.zhimg.com\u002Fv2-38e9a487997d2eea979097fbc9e9e674_r.jpg\"\u002F\u003E\u003C\u002Fnoscript\u003E\u003Cimg src=\"data:image\u002Fsvg+xml;utf8,&lt;svg xmlns=&#39;http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg&#39; width=&#39;523&#39; height=&#39;251&#39;&gt;&lt;\u002Fsvg&gt;\" data-rawwidth=\"523\" data-rawheight=\"251\" data-size=\"normal\" data-default-watermark-src=\"https:\u002F\u002Fpic3.zhimg.com\u002F50\u002Fv2-4d9f5b84e0d1f8aafe616a36b869a012_hd.jpg\" class=\"origin_image zh-lightbox-thumb lazy\" width=\"523\" data-original=\"https:\u002F\u002Fpic1.zhimg.com\u002Fv2-38e9a487997d2eea979097fbc9e9e674_r.jpg\" data-actualsrc=\"https:\u002F\u002Fpic1.zhimg.com\u002F50\u002Fv2-38e9a487997d2eea979097fbc9e9e674_hd.jpg\"\u002F\u003E\u003Cfigcaption\u003E一张地图。边上的数字表示过路费。\u003C\u002Ffigcaption\u003E\u003C\u002Ffigure\u003E\u003Cp\u003E　　这个问题能用DP解决吗？我们先试着记从S到P的最少费用为f(P).\u003Cbr\u002F\u003E　　想要到T，要么经过C，要么经过D。从而\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f%28T%29%3D%5Cmin%E2%81%A1%5C%7Bf%28C%29%2B20%2Cf%28D%29%2B10%5C%7D\" alt=\"f(T)=\\min⁡\\{f(C)+20,f(D)+10\\}\" eeimg=\"1\"\u002F\u003E.\u003C\u002Fp\u003E\u003Cp\u003E　　好像看起来可以DP。现在我们检验刚刚那两个性质：\u003Cbr\u002F\u003E　　- 无后效性：对于点P，一旦f(P)确定，以后就只关心f(P)的值，不关心怎么去的。\u003Cbr\u002F\u003E　　- 最优子结构：对于P，我们当然只关心到P的最小费用，即f(P)。如果我们从S走到T是 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=S+%5Cto+P%5Cto+Q%5Cto+T\" alt=\"S \\to P\\to Q\\to T\" eeimg=\"1\"\u002F\u003E ，那肯定S走到Q的最优路径是 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=S%5Cto+P%5Cto+Q\" alt=\"S\\to P\\to Q\" eeimg=\"1\"\u002F\u003E 。对一条最优的路径而言，从S走到\u003Cb\u003E沿途上所有的点（子问题）\u003C\u002Fb\u003E的最优路径，都是这条大路的一部分。这个问题的最优子结构性质是显然的。\u003C\u002Fp\u003E\u003Cp\u003E　　既然这两个性质都满足，那么本题可以DP。式子明显为：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f%28P%29%3D%5Cmin%E2%81%A1%5C%7Bf%28R%29%2Bw_%7BR%E2%86%92P%7D%5C%7D\" alt=\"f(P)=\\min⁡\\{f(R)+w_{R→P}\\}\" eeimg=\"1\"\u002F\u003E\u003C\u002Fp\u003E\u003Cp\u003E　　其中R为有路通到P的所有的点， \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=w_%7BR%E2%86%92P%7D\" alt=\"w_{R→P}\" eeimg=\"1\"\u002F\u003E 为R到P的过路费。\u003C\u002Fp\u003E\u003Cp\u003E　　代码实现也很简单，拓扑排序即可。\u003C\u002Fp\u003E\u003Ch2\u003E4. 对DP原理的一点讨论\u003C\u002Fh2\u003E\u003Cp\u003E【DP的核心思想】\u003C\u002Fp\u003E\u003Cp\u003E　　DP为什么会快？\u003Cbr\u002F\u003E　　无论是DP还是暴力，我们的算法都是在\u003Cb\u003E可能解空间\u003C\u002Fb\u003E内，寻找\u003Cb\u003E最优解\u003C\u002Fb\u003E。\u003C\u002Fp\u003E\u003Cp\u003E　　来看钞票问题。暴力做法是枚举所有的可能解，这是最大的可能解空间。\u003Cbr\u002F\u003E　　DP是枚举\u003Cb\u003E有希望成为答案的解\u003C\u002Fb\u003E。这个空间比暴力的小得多。\u003C\u002Fp\u003E\u003Cp\u003E　　也就是说：DP自带剪枝。\u003C\u002Fp\u003E\u003Cp\u003E　　DP舍弃了一大堆不可能成为最优解的答案。譬如：\u003Cbr\u002F\u003E　　15 = 5+5+5 被考虑了。\u003Cbr\u002F\u003E　　15 = 5+5+1+1+1+1+1 从来没有考虑过，因为这不可能成为最优解。\u003C\u002Fp\u003E\u003Cp\u003E　　从而我们可以得到DP的核心思想：\u003Cb\u003E尽量缩小可能解空间。\u003C\u002Fb\u003E\u003C\u002Fp\u003E\u003Cp\u003E　　在暴力算法中，可能解空间往往是指数级的大小；如果我们采用DP，那么有可能把解空间的大小降到多项式级。\u003C\u002Fp\u003E\u003Cp\u003E　　一般来说，解空间越小，寻找解就越快。这样就完成了优化。\u003C\u002Fp\u003E\u003Cp class=\"ztext-empty-paragraph\"\u003E\u003Cbr\u002F\u003E\u003C\u002Fp\u003E\u003Cp\u003E【DP的操作过程】\u003C\u002Fp\u003E\u003Cp\u003E　　一言以蔽之：\u003Cb\u003E大事化小，小事化了。\u003C\u002Fb\u003E\u003C\u002Fp\u003E\u003Cp\u003E　　将一个大问题转化成几个小问题；\u003Cbr\u002F\u003E　　求解小问题；\u003Cbr\u002F\u003E　　推出大问题的解。\u003C\u002Fp\u003E\u003Cp\u003E【如何设计DP算法】\u003C\u002Fp\u003E\u003Cp\u003E　　下面介绍比较通用的设计DP算法的步骤。\u003C\u002Fp\u003E\u003Cp\u003E　　首先，把我们面对的\u003Cb\u003E局面\u003C\u002Fb\u003E表示为x。这一步称为\u003Cb\u003E设计状态\u003C\u002Fb\u003E。\u003Cbr\u002F\u003E　　对于状态x，记我们要求出的答案(e.g. 最小费用)为f(x).我们的目标是求出f(T).\u003Cbr\u002F\u003E\u003Cb\u003E找出f(x)与哪些局面有关（记为p）\u003C\u002Fb\u003E，写出一个式子（称为\u003Cb\u003E状态转移方程\u003C\u002Fb\u003E），通过f(p)来推出f(x).\u003C\u002Fp\u003E\u003Cp\u003E【DP三连】\u003C\u002Fp\u003E\u003Cp\u003E　　设计DP算法，往往可以遵循DP三连：\u003C\u002Fp\u003E\u003Cp\u003E　　我是谁？  ——设计状态，表示局面\u003Cbr\u002F\u003E　　我从哪里来？\u003Cbr\u002F\u003E　　我要到哪里去？  ——设计转移\u003C\u002Fp\u003E\u003Cp\u003E　　设计状态是DP的基础。接下来的设计转移，有两种方式：一种是考虑我从哪里来（本文之前提到的两个例子，都是在考虑“我从哪里来”）；另一种是考虑我到哪里去，这常见于求出f(x)之后，\u003Cb\u003E更新能从x走到的一些解\u003C\u002Fb\u003E。这种DP也是不少的，我们以后会遇到。\u003C\u002Fp\u003E\u003Cp\u003E　　总而言之，“我从哪里来”和“我要到哪里去”只需要考虑清楚其中一个，就能设计出状态转移方程，从而写代码求解问题。前者又称pull型的转移，后者又称push型的转移。（这两个词是 \u003Ca class=\"member_mention\" href=\"https:\u002F\u002Fwww.zhihu.com\u002Fpeople\u002F31585b1079521ef37391925a95a5d4ba\" data-hash=\"31585b1079521ef37391925a95a5d4ba\" data-hovercard=\"p$b$31585b1079521ef37391925a95a5d4ba\"\u003E@阮止雨\u003C\u002Fa\u003E 妹妹告诉我的，不知道源出处在哪）\u003C\u002Fp\u003E\u003Cblockquote\u003E思考题：如何把钞票问题的代码改写成“我到哪里去”的形式？\u003Cbr\u002F\u003E提示：求出f(x)之后，更新f(x+1),f(x+5),f(x+11).\u003C\u002Fblockquote\u003E\u003Ch2\u003E5. 例题：最长上升子序列\u003C\u002Fh2\u003E\u003Cp\u003E　　扯了这么多形而上的内容，还是做一道例题吧。\u003C\u002Fp\u003E\u003Cp\u003E　　最长上升子序列（LIS）问题：给定长度为n的序列a，从a中抽取出一个子序列，这个子序列需要单调递增。问最长的上升子序列（LIS）的长度。\u003Cbr\u002F\u003E　　e.g. 1,5,3,4,6,9,7,8的LIS为1,3,4,6,7,8，长度为6。\u003C\u002Fp\u003E\u003Cp class=\"ztext-empty-paragraph\"\u003E\u003Cbr\u002F\u003E\u003C\u002Fp\u003E\u003Cp\u003E　　如何设计状态（我是谁）？\u003C\u002Fp\u003E\u003Cp\u003E　　我们记 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f%28x%29\" alt=\"f(x)\" eeimg=\"1\"\u002F\u003E 为以 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=a_x\" alt=\"a_x\" eeimg=\"1\"\u002F\u003E 结尾的LIS长度，那么答案就是 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=%5Cmax%5C%7Bf%28x%29%5C%7D\" alt=\"\\max\\{f(x)\\}\" eeimg=\"1\"\u002F\u003E .\u003C\u002Fp\u003E\u003Cp class=\"ztext-empty-paragraph\"\u003E\u003Cbr\u002F\u003E\u003C\u002Fp\u003E\u003Cp\u003E　　状态x从哪里推过来（我从哪里来）？\u003C\u002Fp\u003E\u003Cp\u003E　　考虑比x小的每一个p：如果 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=a_x%3Ea_p\" alt=\"a_x&gt;a_p\" eeimg=\"1\"\u002F\u003E ，那么f(x)可以取f(p)+1.\u003Cbr\u002F\u003E　　解释：我们把 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=a_x\" alt=\"a_x\" eeimg=\"1\"\u002F\u003E 接在 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=a_p\" alt=\"a_p\" eeimg=\"1\"\u002F\u003E 的后面，肯定能构造一个以 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=a_x\" alt=\"a_x\" eeimg=\"1\"\u002F\u003E 结尾的上升子序列，长度比以 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=a_p\" alt=\"a_p\" eeimg=\"1\"\u002F\u003E 结尾的LIS大1.那么，我们可以写出状态转移方程了：\u003C\u002Fp\u003E\u003Cp\u003E\u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=f%28x%29%3D%5Cmax_%7Bp%3Cx+%2C+a_p%3Ca_x+%7D%E2%81%A1%5C%7Bf%28p%29%5C%7D%2B1\" alt=\"f(x)=\\max_{p&lt;x , a_p&lt;a_x }⁡\\{f(p)\\}+1\" eeimg=\"1\"\u002F\u003E\u003C\u002Fp\u003E\u003Cp\u003E　　至此解决问题。两层for循环，复杂度 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=O%28n%5E2%29\" alt=\"O(n^2)\" eeimg=\"1\"\u002F\u003E .\u003C\u002Fp\u003E\u003Cfigure data-size=\"normal\"\u003E\u003Cnoscript\u003E\u003Cimg src=\"https:\u002F\u002Fpic3.zhimg.com\u002F50\u002Fv2-73ea19922aaac11c15dff9146a5c5b41_hd.jpg\" data-rawwidth=\"883\" data-rawheight=\"455\" data-size=\"normal\" data-caption=\"\" data-default-watermark-src=\"https:\u002F\u002Fpic2.zhimg.com\u002F50\u002Fv2-ce272e9f786ef6377691df709264711a_hd.jpg\" class=\"origin_image zh-lightbox-thumb\" width=\"883\" data-original=\"https:\u002F\u002Fpic3.zhimg.com\u002Fv2-73ea19922aaac11c15dff9146a5c5b41_r.jpg\"\u002F\u003E\u003C\u002Fnoscript\u003E\u003Cimg src=\"data:image\u002Fsvg+xml;utf8,&lt;svg xmlns=&#39;http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg&#39; width=&#39;883&#39; height=&#39;455&#39;&gt;&lt;\u002Fsvg&gt;\" data-rawwidth=\"883\" data-rawheight=\"455\" data-size=\"normal\" data-caption=\"\" data-default-watermark-src=\"https:\u002F\u002Fpic2.zhimg.com\u002F50\u002Fv2-ce272e9f786ef6377691df709264711a_hd.jpg\" class=\"origin_image zh-lightbox-thumb lazy\" width=\"883\" data-original=\"https:\u002F\u002Fpic3.zhimg.com\u002Fv2-73ea19922aaac11c15dff9146a5c5b41_r.jpg\" data-actualsrc=\"https:\u002F\u002Fpic3.zhimg.com\u002F50\u002Fv2-73ea19922aaac11c15dff9146a5c5b41_hd.jpg\"\u002F\u003E\u003C\u002Ffigure\u003E\u003Cp\u003E　　从这三个例题中可以看出，DP是一种思想，一种“大事化小，小事化了”的思想。带着这种思想，DP将会成为我们解决问题的利器。\u003C\u002Fp\u003E\u003Cp\u003E　　最后，我们一起念一遍DP三连吧——我是谁？我从哪里来？我要到哪里去？\u003C\u002Fp\u003E\u003Ch2\u003E6. 习题\u003C\u002Fh2\u003E\u003Cp\u003E如果读者有兴趣，可以试着完成下面几个习题：\u003C\u002Fp\u003E\u003Cp\u003E一、请采取一些优化手段，以 \u003Cimg src=\"https:\u002F\u002Fwww.zhihu.com\u002Fequation?tex=O%28n%5Clog+n%29\" alt=\"O(n\\log n)\" eeimg=\"1\"\u002F\u003E 的复杂度解决LIS问题。\u003C\u002Fp\u003E\u003Cp\u003E提示：可以参考这篇博客 \u003Ca href=\"https:\u002F\u002Flink.zhihu.com\u002F?target=https%3A\u002F\u002Fpks-loving.blog.luogu.org\u002Fjunior-dynamic-programming-dong-tai-gui-hua-chu-bu-ge-zhong-zi-xu-lie\" class=\" wrap external\" target=\"_blank\" rel=\"nofollow noreferrer\"\u003EJunior Dynamic Programming--动态规划初步·各种子序列问题\u003C\u002Fa\u003E\u003C\u002Fp\u003E\u003Cp\u003E二、“按顺序递推”和“记忆化搜索”是实现DP的两种方式。请查阅资料，简单描述“记忆化搜索”是什么。并采用记忆化搜索写出钞票问题的代码，然后完成\u003Ca href=\"https:\u002F\u002Flink.zhihu.com\u002F?target=https%3A\u002F\u002Fwww.luogu.org\u002Fproblemnew\u002Fshow\u002FP1541\" class=\" wrap external\" target=\"_blank\" rel=\"nofollow noreferrer\"\u003EP1541 乌龟棋 - 洛谷\u003C\u002Fa\u003E 。\u003C\u002Fp\u003E\u003Cp\u003E三、01背包问题是一种常见的DP模型。请完成\u003Ca href=\"https:\u002F\u002Flink.zhihu.com\u002F?target=https%3A\u002F\u002Fwww.luogu.org\u002Fproblemnew\u002Fshow\u002FP1048\" class=\" wrap external\" target=\"_blank\" rel=\"nofollow noreferrer\"\u003EP1048 采药 - 洛谷\u003C\u002Fa\u003E。\u003C\u002Fp\u003E\u003Cp class=\"ztext-empty-paragraph\"\u003E\u003Cbr\u002F\u003E\u003C\u002Fp\u003E\u003Cp\u003E谢谢您看完本文 ⁄(⁄ ⁄ ⁄ω⁄ ⁄ ⁄)⁄\u003C\u002Fp\u003E\u003Cp\u003E2019.3.3\u003C\u002Fp\u003E","editableContent":"","excerpt":"0. intro 很有意思的问题。以往见过许多教材，对动态规划（DP）的引入属于“奉天承运，皇帝诏曰”式：不给出一点引入，见面即拿出一大堆公式吓人；学生则死啃书本，然后突然顿悟。针对入门者的教材不应该是这样的。恰好我给入门者讲过四次DP入门，迭代出了一套比较靠谱的教学方法，所以今天跑过来献丑。 现在，我们试着自己来一步步“重新发明”DP。 1. 从一个生活问题谈起 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